Finite Group Databasebeta

← #637407

Subgroups — #637407

Order 512.

3106 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C211yesmaximal, characteristic
2C2×#2297C211yesmaximal, characteristic
3#16432C214yesmaximal, complemented
4#16309C214yesmaximal, complemented
5#15226C212yesmaximal, complemented
6#15205C214yesmaximal, complemented
7#15186C212yesmaximal, complemented
8#15180C212yesmaximal, complemented
9#15147C212yesmaximal, complemented
10#14702C211yesmaximal, characteristic, complemented
11#14304C214yesmaximal, complemented
12#14138C211yesmaximal, characteristic, complemented
13#14129C211yesmaximal, characteristic, complemented
14#13937C211yesmaximal, characteristic
15C4×#385C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2432—28no—
23C2×#397C2214yes—
24C2×#394C2214yes—
25#2297—28no—
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#386C2214yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44#1445C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2212yes—
55#1440C2214yes—
56#1440C2212yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1438C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2212yes—
67#1436C2212yes—
68#1436C2212yes—
69#1436C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—44no—
80C22×#41—44no—
81C22×#41D418yes—
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—44no—
91C2×#124—24no—
92C2×#124—44no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×#117—24no—
97C2×#117—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C42—24no—
119C22×C42—44no—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#397—216no—
128#394—216no—
129#391—24no—
130#391—28no—
131#391—44no—
132#391—44no—
133#391—44no—
134#390—24no—
135#390—28no—
136#390—44no—
137#389—44no—
138#386—216no—
139#385—216no—
140#385—28no—
141#380—24no—
142C4×C4⋊C4—44no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—24no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2412yes—
173C25C2411yescharacteristic
174C2×#41—232no—
175C2×#41—416no—
176C2×#41—416no—
177C2×#41—432no—
178C2×#41—432no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8C2×D414yes—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4C2×D414yes—
198C22×D4C2×D418yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4C2×D414yes—
213C23×C4C2×D418yes—
214C23×C4#4114yes—
215C23×C4#4114yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C4⋊Q8—416no—
230#124—416no—
231C22⋊Q8—416no—
232#117—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4#4118yescomplemented
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—44no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—416no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24C22×D414yes—
346C24C22×D412yes—
347C24C2×#4112yes—
348C24C2×#4112yes—
349C24C2511yescharacteristic
350#41—4128no—
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×D4—216no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—48no—
370C2×D4—48no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—216no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4#123116yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)14yes—
468C22×C42−1+414yes—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—416no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—88no—
477C22⋊C4—28no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—88no—
487C22⋊C4—84no—
488C22⋊C4—88no—
489C42—44no—
490C42—44no—
491C42—48no—
492C42—48no—
493C42—44no—
494C42—44no—
495C42—48no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23C2×#12314yes—
538C23C23×D411yescharacteristic
539C23C22×#4111yescharacteristic
540C23C22×#4111yescharacteristic
541C23C22×#4112yes—
542C23C2×H(5,𝔽2)12yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×2−1+412yes—
545C23#58512yes—
546C23#58512yes—
547Q8—416no—
548Q8—88no—
549D4—432no—
550D4—416no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—232no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—432no—
572C2×C4—432no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4C2×#12314yes—
630C2×C4C2×#12314yes—
631C2×C4#53414yes—
632C2×C4#53414yes—
633C2×C4#53818yes—
634C2×C4#53818yes—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—48no—
645C22—44no—
646C22—48no—
647C22—44no—
648C22—44no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22C22×#12311yescharacteristic
654C22#344411yescharacteristic
655C22C2×#53411yescharacteristic
656C22C2×#53411yescharacteristic
657C22C2×#53812yes—
658C22C2×#53812yes—
659C22#345511yescharacteristic
660C22#346112yes—
661C22D4×#4112yes—
662C22#348012yes—
663C22#348912yes—
664C22#350512yes—
665C22#350911yescharacteristic
666C22#351011yescharacteristic
667C22#351112yes—
668C22#351712yes—
669C22#352012yes—
670C22#354012yes—
671C22#354112yes—
672C22#355412yes—
673C22#356212yes—
674C4—24no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—84no—
685C4—82no—
686C4#262218yes—
687C2—44no—
688C2—84no—
689C2C2×#262211yescharacteristic
690C2#3474411yescharacteristic
691C2#3503612yes—
692C2#3530711yescharacteristic
693C2#3749912yes—
694C2#3986012yes—
695C2#3988912yes—
696C2#4003112yes—
697C2#4003312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #16432C2semidirect
4. #16309C2semidirect
5. #15226C2semidirect
6. #15205C2semidirect
7. #15186C2semidirect
8. #15180C2semidirect
9. #15147C2semidirect
10. #14702C2semidirect
11. #14304C2semidirect
12. #14138C2semidirect
13. #14129C2semidirect
15. C4×#385C2semidirect
256. C2×C4⋊C4#41semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#24322
C2×#22972
#164322
#163092
#152262
#152052
#151862
#151802
#151472
#147022
#143042
#141382
#141292
#139372
C4×#3852